\(\def\N{\mathbb{N}}\) \(\def\Z{\mathbb{Z}}\) \(\def\Q{\mathbb{Q}}\) \(\def\R{\mathbb{R}}\) \(\def\C{\mathbb{C}}\) \(\def\H{\mathbb{H}}\) \(\def\6{\partial}\) \(\DeclareMathOperator\Res{Res}\) \(\DeclareMathOperator\M{M}\) \(\DeclareMathOperator\ord{ord}\) \(\DeclareMathOperator\const{const}\) \(\DeclareMathOperator{\arccosh}{arccosh}\) \(\DeclareMathOperator{\arcsinh}{arcsinh}\) \(\DeclareMathOperator\id{id}\) \(\DeclareMathOperator\rk{rk}\) \(\DeclareMathOperator\tr{tr}\) \(\def\pt{\mathrm{pt}}\) \(\DeclareMathOperator\colim{colim}\) \(\DeclareMathOperator\Hom{Hom}\) \(\DeclareMathOperator\End{End}\) \(\DeclareMathOperator\Aut{Aut}\) \(\let\Im\relax\DeclareMathOperator\Im{Im}\) \(\let\Re\relax\DeclareMathOperator\Re{Re}\) \(\DeclareMathOperator\Ker{Ker}\) \(\DeclareMathOperator\Coker{Coker}\) \(\DeclareMathOperator\Map{Map}\) \(\def\GL{\mathrm{GL}}\) \(\def\SL{\mathrm{SL}}\) \(\def\O{\mathrm{O}}\) \(\def\SO{\mathrm{SO}}\) \(\def\Spin{\mathrm{Spin}}\) \(\def\U{\mathrm{U}}\) \(\def\SU{\mathrm{SU}}\) \(\def\g{{\mathfrak g}}\) \(\def\h{{\mathfrak h}}\) \(\def\gl{{\mathfrak{gl}}}\) \(\def\sl{{\mathfrak{sl}}}\) \(\def\sp{{\mathfrak{sp}}}\) \(\def\so{{\mathfrak{so}}}\) \(\def\spin{{\mathfrak{spin}}}\) \(\def\u{{\mathfrak u}}\) \(\def\su{{\mathfrak{su}}}\) \(\def\cA{\mathcal{A}}\) \(\def\cB{\mathcal{B}}\) \(\def\cC{\mathcal{C}}\) \(\def\cD{\mathcal{D}}\) \(\def\cE{\mathcal{E}}\) \(\def\cF{\mathcal{F}}\) \(\def\cG{\mathcal{G}}\) \(\def\cH{\mathcal{H}}\) \(\def\cI{\mathcal{I}}\) \(\def\cJ{\mathcal{J}}\) \(\def\cK{\mathcal{K}}\) \(\def\cL{\mathcal{L}}\) \(\def\cM{\mathcal{M}}\) \(\def\cN{\mathcal{N}}\) \(\def\cO{\mathcal{O}}\) \(\def\cP{\mathcal{P}}\) \(\def\cQ{\mathcal{Q}}\) \(\def\cR{\mathcal{R}}\) \(\def\cS{\mathcal{S}}\) \(\def\cT{\mathcal{T}}\) \(\def\cU{\mathcal{U}}\) \(\def\cV{\mathcal{V}}\) \(\def\cW{\mathcal{W}}\) \(\def\cX{\mathcal{X}}\) \(\def\cY{\mathcal{Y}}\) \(\def\cZ{\mathcal{Z}}\) \(\def\al{\alpha}\) \(\def\be{\beta}\) \(\def\ga{\gamma}\) \(\def\de{\delta}\) \(\def\ep{\epsilon}\) \(\def\ze{\zeta}\) \(\def\th{\theta}\) \(\def\io{\iota}\) \(\def\ka{\kappa}\) \(\def\la{\lambda}\) \(\def\si{\sigma}\) \(\def\up{\upsilon}\) \(\def\vp{\varphi}\) \(\def\om{\omega}\) \(\def\De{\Delta}\) \(\def\Ka{{\rm K}}\) \(\def\La{\Lambda}\) \(\def\Om{\Omega}\) \(\def\Ga{\Gamma}\) \(\def\Si{\Sigma}\) \(\def\Th{\Theta}\) \(\def\Up{\Upsilon}\) \(\def\Chi{{\rm X}}\) \(\def\Tau{{T}}\) \(\def\Nu{{\rm N}}\) \(\def\op{\oplus}\) \(\def\ot{\otimes}\) \(\def\t{\times}\) \(\def\bt{\boxtimes}\) \(\def\bu{\bullet}\) \(\def\iy{\infty}\) \(\def\longra{\longrightarrow}\) \(\def\an#1{\langle #1 \rangle}\) \(\def\ban#1{\bigl\langle #1 \bigr\rangle}\) \(\def\llbracket{{\normalsize\unicode{x27E6}}} \def\rrbracket{{\normalsize\unicode{x27E7}}} \) \(\def\lb{\llbracket}\) \(\def\rb{\rrbracket}\) \(\def\ul{\underline}\) \(\def\ol{\overline}\)

1  Geometry of the Complex plane

This section is a brief reminder of Sections 3 and 4 of MA1006 Algebra.

Definition 1.1 The complex numbers \(\C\) are the set of all pairs \(z=(x,y)\in\R^2\) of real numbers with the addition

\[ z_1+z_2=(x_1+x_2,y_1+y_2) \tag{1.1}\]

and the multiplication

\[ z_1\cdot z_2=(x_1x_2-y_1y_2,x_1y_2+x_2y_1), \tag{1.2}\]

where \(z_1=(x_1,y_1)\) and \(z_2=(x_2,y_2).\) We call \(x=\Re(z)\) the real part and \(y=\Im(z)\) the imaginary part of \(z.\)

From Section 3.8 in MA1006 Algebra we recall:

Proposition 1.1 The complex numbers are a field.

Remark 1.1.

We view the real numbers as a subset of \(\C\) by identifying \(x\in\R\) with \((x,0)\in\C.\) The imaginary unit is \(i=(0,1).\) With these conventions, a calculation using Equation 1.2 shows that

\[ z=x+iy. \tag{1.3}\]

Using this notation, we can manipulate complex numbers in the same way as real numbers, keeping in mind the identity

\[ i^2=-1. \tag{1.4}\]

Of course, \(\C\) is a one-dimensional vector space over itself. Restricting the scalar multiplication to \(\R\) makes \(\C\) into a vector space over \(\R,\) isomorphic to \(\R^2,\) of dimension two with standard basis \(1,i\in\C.\)

Proposition 1.2  

  1. In the standard basis, every \(A=\begin{pmatrix}a&b \\ c&d\end{pmatrix} \in\M_{2\t2}(\R)\) corresponds uniquely to an \(\R\)-linear map \(T_A\colon\C\to\C,\) namely \[ T_A(x+iy)=(ax+by)+i(cx+dy). \tag{1.5}\]

  2. \(T_A\) is \(\C\)-linear \(\iff\) \(a=d\) and \(b=-c.\) In this case, \[ T_A(z)=\al\cdot z,\qquad \al=a+ic. \tag{1.6}\]

Proof.

  1. This is a recap from linear algebra. An \(\R\)-linear map \(T\colon\C\to\C\) is uniquely determined by the image of the basis vectors \(T(1),\) \(T(i)\) which, conversely, may be prescribed arbitrarily. If we write \(T(1)=a+ic,\) \(T(i)=b+id,\) then the linear transform \(T\) is described entirely by \(a,b,c,d\in\R.\) The expression Equation 1.5 is obtained by expanding the left hand side by \(\R\)-linearity.

  2. In the same way, \(\C\)-linear maps correspond to \(\al\in\M_{1\t1}(\C)\) as in Equation 1.6. An \(\R\)-linear map \(T_A\) is \(\C\)-linear \(\iff\) \(T_A(i)=T_A(i1)=iT_A(1)\) \(\iff\)

    \[\begin{pmatrix}b \\ d \end{pmatrix} =T_A(i)=iT_A(1)=i\begin{pmatrix}a \\ c \end{pmatrix} =\begin{pmatrix}-c \\ a \end{pmatrix}.\]

    In this case, \(T_A(x+iy)=(ax-cy)+i(cx+ay)=\al(x+iy).\)

Definition 1.2 The conjugate of \(z\in\C\) is the complex number \[\ol{z}=(x,-y),\]and the modulus (also called absolute value) is\[|z|=\sqrt{x^2+y^2}\geqslant0.\]

Proposition 1.3 The following formulas hold for \(z,w\in \C:\)

\[\begin{align} \ol{z\cdot w}&=\ol{z}\cdot\ol{w} &\ol{z+w}&=\ol{z}+\ol{w}\\ \ol{\ol{z}}&=z &\ol{i}=-i&,\ \ol{1}=1\\ z\cdot\ol{z}&=|z|^2 &|z\cdot w|&=|z|\cdot|w|\\ \Re(z)&=\frac{z+\ol{z}}{2} &\Im(z)&=\frac{z-\ol{z}}{2i}\\ z^{-1}&=\frac{\ol{z}}{|z|^2}\quad\text{if $z\neq0$} \end{align}\]

Proposition 1.4 The following inequalities hold for \(z,w\in\C:\)

\[|z+w|\leqslant|z|+|w| \qquad |z-w|\geqslant\bigl||z|-|w|\bigr| \tag{1.7}\]

\[|\Re(z)|\leqslant|z| \qquad |\Im(z)|\leqslant|z| \tag{1.8}\]

Thinking of complex numbers as points in the plane, we can use polar coordinates to represent them (see Figure 1.1).

Figure 1.1: Polar coordinates

Proposition 1.5 For every non-zero complex number \(z=(x,y)\) there is an argument \(\th\in\R\) and a radius \(r=|z|>0\) such

\[x=r\cos(\th), y=r\sin(\th). \tag{1.9}\]

This representation is unique up to replacing \(\th\) by \(\th+2\pi k\) for any \(k\in\Z.\)

Proof. (omitted)

Since \(x^2+y^2=r^2(\cos(\th)^2+\sin(\th)^2)=r^2,\) the radius must be \(r=|z|.\) Define the complex number \(w=r^{-1}z\) and write \(w=u+iv\) for its real and imaginary parts.

We will prove the existence of \(\th\in\R\) with \(u=\cos(\th),\) \(v=\sin(\th).\) This also proves the existence of a representation Equation 1.9, by multiplying by \(r.\) Since \(u^2+v^2=|w|^2=r^{-2}|z|^2=1,\) we know \(|u|\leqslant1,\) \(|v|\leqslant1.\) Recall that \(\cos\colon[0,\pi]\to[-1,1]\) and \(\sin\colon[-\pi/2,\pi/2]\to[-1,1]\) are bijections. Hence \[\begin{align*} u&=\cos(\al) &&\text{ for some }\al\in[0,\pi], \\ v&=\sin(\be) &&\text{ for some }\be\in[-\pi/2,\pi/2]. \end{align*}\] As \(\sin(\be)^2=v^2=1-u^2=1-\cos(\al)^2=\sin(\al)^2,\) we have \(\sin(\al)=\pm\sin(\be)=\sin(\pm\be).\) To produce the correct \(\th,\) we distinguish two cases.

Case 1
\(\al\in[0,\pi/2].\) Then \(\al=\pm\be\) by the injectivity of the sine function on the interval \([-\pi/2,\pi/2].\) Setting \(\th=\pm\al=\be,\) we find that \(u=\cos(\th)\) and \(v=\sin(\th),\) as required.
Case 2
\(\al\in[\pi/2,\pi].\) Then \(\pi-\al, \be\in[-\pi/2,\pi/2]\) and \(\sin(\pi-\al)=\sin(\al)=\pm\sin(\be),\) so \(\pi-\al=\pm\be\) by injectivity. Setting \(\th=\pm\al=\pm\pi-\be,\) we find \(u=\cos(\th), v=\sin(\th),\) using trigonometric identities.

This completes the existence part of the proof. For uniqueness, we already know that \(r=|z|>0\) is unique, so it remains to consider \[\begin{align*} x&=r\cos(\th_1)=r\cos(\th_2), &y&=r\sin(\th_1)=r\sin(\th_2). \end{align*}\] To translate the situation into an interval that we understand, pick \(k_1, k_2\in \Z\) so that \(\th_1 + 2\pi k_1, \th_2 + 2\pi k_2 \in [-\pi,\pi).\) Then \[\cos(\th_1+2\pi k_1)=\cos(\th_1)=\cos(\th_2)=\cos(\th_2+2\pi k_2).\]

Using the injectivity of the cosine function and considering cases as above, we find that \(\th_1+2\pi k_1 = \pm(\th_2+2\pi k_2).\) If the sign is ‘\(+\)’ we get \(\th_1-\th_2=2\pi(k_2-k_1)\) and we are done, so suppose \(\th_1+2\pi k_1 = -(\th_2+2\pi k_2).\) Then \[\sin(\th_1)=\sin(\th_2)=\sin(\th_2+2\pi k_1)=-\sin(\th_1+2\pi k_1)=-\sin(\th_1)\]

implies \(\sin(\th_1)=0.\) Therefore \(\th_1\) is a multiple of \(2\pi,\) which implies that \(\th_1+2\pi k_1 = -(\th_2+2\pi k_2)=0\) since these numbers were chosen in \([-\pi,\pi)\) and we have \(2\pi\Z\cap[-\pi,\pi)=\{0\}.\) Hence \(\th_1-\th_2=2\pi(k_2-k_1).\)

To get around the non-uniqueness of the argument in polar coordinates, we restrict \(\th\) to lie in a half-open interval of length \(2\pi.\) Here is the most common convention.

Definition 1.3 The principal argument of a non-zero \(z\in\C\) is the unique \(\th\in(-\pi,\pi]\) such that Equation 1.9 holds, and we write \(\arg(z)=\th.\)

Definition 1.4 The value of the exponential function at the complex number \(z=x+i\th,\) where \(x,\th\in\R,\) is defined as \[e^{x+i\th}=e^x\bigl(\cos(\th)+i\sin(\th)\bigr). \tag{1.10}\]

Proposition 1.5 implies that every complex number can be represented in polar form\[z=re^{i\th}. \tag{1.11}\]

The addition of complex numbers is the usual addition of vectors in \(\R^2.\) To visualize multiplication, the polar form is useful. Combining Equation 1.2 and Equation 1.10, we find that \[\begin{align*} e^{i\th_1}\cdot e^{i\th_2}=&\cos(\th_1)\cos(\th_2)-\sin(\th_1)\sin(\th_2)\\ &+i\bigl[\cos(\th_1)\sin(\th_2)+\cos(\th_2)\sin(\th_1)\bigr] \\ =&\cos(\th_1+\th_2)+i\sin(\th_1+\th_2)=e^{i(\th_1+\th_2)}. \end{align*}\] Here we have used the trigonometric addition formulas. Hence

\[z_1 = r_1e^{i\th_1}, z_2 = r_2e^{i\th_2} \implies z_1z_2 = r_1r_2e^{i(\th_1+\th_2)}. \tag{1.12}\]

Complex multiplication adds the angles and multiplies the radii.

Proposition 1.6 \(e^{z_1}\cdot e^{z_2}=e^{z_1+z_2}\) for all \(z_1, z_2\in\C.\) Moreover, we have \((e^z)^n=e^{nz}\) for all \(z\in\C,\) \(n\in\Z.\)

Proof.

The first assertion follows from Equation 1.12 combined with the identity \(e^{x_1}e^{x_2}=e^{x_1+x_2}\) for \(x_1, x_2\in \R\) from MA1005 Calculus. The second claim follows from this by induction.

The polar form can be applied to the construction of \(n\)th roots.

For example, the nth root of unity is \(\ze_n=e^{i\frac{2\pi}{n}}\) and satisfies \[(\ze_n)^n=(e^{i\frac{2\pi}{n}})^n=e^{2\pi i}=1.\]

Proposition 1.7 Every complex number \(z\neq0\) has an \(n\)th root \(w\) satisfying \(w^n=z.\) If \(w\) is an \(n\)th root of \(z,\) the set of all \(n\)th roots of \(z\) is \[\bigl\{w, \ze_n\cdot w, \ze_n^2\cdot w,\ldots, \ze_n^{n-1}\cdot w\bigr\}. \]

Proof.

Write \(z=re^{i\th}\) and \(w=se^{i\varphi}\) for \(\th,\varphi\in[0,2\pi)\) and \(r,s>0.\) By the uniqueness of the polar form, the equation \(w^n=z\) is equivalent to \(s^n=r\) and \(n\varphi=\th+2\pi k\) for some \(k\in\Z.\) Of course, \(s=\sqrt[n]{r}\) is the unique positive \(n\)th root from MA1005 Calculus. From \(0\leqslant n\varphi<2\pi n\) and \(0<-\th\leqslant2\pi\) we get \[0<k=\frac{n\varphi-\th}{2\pi}<n.\]

Since \(k\) is an integer, this implies \(k=0,\ldots,n-1\) and hence \(\varphi=\frac{\th+2\pi k}{n}\) for such \(k\) are the only possible solutions for \(\varphi.\) In summary, \[w_k = \sqrt[n]{r}e^{i\varphi}=\sqrt[n]{r}e^{i\frac{\th+2\pi k}{n}}=\ze_n^k\cdot w_0, k=0,\ldots,n-1,\]

are all the possible \(n\)th roots of \(z.\)

Questions for further discussion

  • The complex numbers are obtained by ‘adjoining’ a symbol \(i\) with \(i^2=-1.\) If instead we would have adjoined a different symbol \(\ep\) with \(\ep^2=-1,\) would the set of elements \(x+\ep y\) still define a field?
  • The real numbers have a total order ‘\(\leqslant\)’. Why doesn’t it make sense to extend this definition to the complex numbers?
  • Describe geometrically the set \(R_n=\{1,\ze_n,\ldots,(\ze_n)^{n-1}\}\) of \(n\)th roots of unity. Find a connection between \(R_n\) and the cyclic group \(C_n=\{\ol{0},\ldots,\ol{n-1}\}\) of integers modulo \(n\) from MX3020 Group Theory.

1.1 Exercises

Note

This problem sheet is intended as a recap and contains more problems than can be discussed during the tutorials.

Exercise 1.1

Verify \[z = x+iy\] and \[i^2 =-1\] straight from the definition Equation 1.2.

Exercise 1.2

How many real solutions \(x\) does \(x^2+1=0\) have? Show that the polynomial equation \(z^2+1=0\) has exactly two solutions \(z\in\C.\)

Exercise 1.3

Give examples of complex numbers \(z,w\neq0\) such that \(z^2+w^2=0.\)

Exercise 1.4

Sketch the position of the complex numbers \(i, 1+i, \frac{3+2i}{4}\) in the plane.

Exercise 1.5

Express the following complex numbers \(z\) in the form \(x+iy\) with \(x,y\in\R.\) \[(1+i)^{20},\ (5+3i)(1+2i),\ (1-i)(2+3i),\ (1-i)i(1+i),\ \frac{2+i}{1-i}\]

Exercise 1.6

Express the following complex numbers \(z\) in the form \(x+iy\) with \(x,y\in\R.\) \[1/i,\ \frac{1}{1+i},\ \frac{3+i}{3-i} \]

Exercise 1.7

Find the modulus and the conjugate of the following complex numbers. \[2+i,\ i,\ 5-3i,\ \frac{1+i}{2+i} \]

Exercise 1.8

Describe the sets \(A=\{z\in\C\mid\Im(z)>0\},\) \(B=\{z\in\C\mid\Re(z)\leqslant1\},\) \(C=\{z\in\C\mid \Re((1+i)z)=0\},\) and \(A\cap B\) geometrically.

Exercise 1.9

Describe the set \(D=\{z\in\C \mid z\cdot\overline{z}=1\}\) geometrically.

Hint: Write \(z=re^{i\th}\) in polar form.

Exercise 1.10

Draw all nine sets described by the following conditions on the complex number \(z.\) \[\begin{align*} |z|&=1,&|z|&<1,&1<&|z|<2,\\ |1+z|&>1,&|2-z|&<2,&3<&|z+i|<4,\\ |z-1|&<|z+1|,&|z|&=|z+1|,&|z-1|&=|z+i|. \end{align*}\]

Exercise 1.11

Let \(S=\{x+iy\in \C \mid 0\leqslant x,y \leqslant 1\}\). Draw \(S\) and the sets \[\begin{align*} A&=\{2z \mid z\in S\}, &B&=\{\ol{z} \mid z\in S\},\\ C&=\{-z \mid z\in S\}, &D&=\{z^2 \mid z\in S\}. \end{align*}\]

Exercise 1.12

Let \(D=\{z\in\C \mid |z|<1\}\) be the unit disk. Draw the sets \[A=\{2z \mid z\in D\},\quad B=\{z^2 \mid z\in D\},\quad C=\{|z| \mid z\in D\}.\]

Exercise 1.13

Show that \(i=e^{i\pi/2}\) and \(-1=e^{i\pi}.\)

Exercise 1.14

Express the following complex numbers \(z\) in the form \(x+iy\) with \(x,y\in\R.\) \[e^{i\pi/4},\ e^{i\pi},\ e^{i\frac{2\pi}{3}} \]

Exercise 1.15

Write each of the following complex numbers in polar form \(re^{i\theta}\) with \(r>0\) and \(-\pi<\theta\leqslant\pi.\) \[i, \quad -1, \quad -i, \quad 1+i, \quad 1-i, \quad i-1, \quad \frac{1}{2}+i\frac{\sqrt{3}}{2}\] Draw each of these numbers in the complex plane.

Exercise 1.16

Calculate \(i^{2021}\) and \((1+i)^{20}.\)

Exercise 1.17

Solve the equation \((1 - i)^n - 2075 = 2021\) and find \(n\in\N.\)

Exercise 1.18

Prove that for \(z\in\R\) we have

\[\begin{align*} \cos(z)&=\frac{e^{iz}+e^{-iz}}{2}, &\sin(z)&=\frac{e^{iz}-e^{-iz}}{2i}. \end{align*}\]

Use these equations to extend the definition of the functions \(\cos(z), \sin(z)\) to complex arguments \(z\in\C.\) Find \(z\in\C\) with \(\sin(z)=2.\)

Hint: Put \(w=e^{iz}\) and reduce to a quadratic equation.

Exercise 1.19

Prove the following statements.

  1. \(\ol{zw}=\ol{z}\cdot\ol{w}\) for all \(z,w\in\C\)
  2. \(\ol{z_1z_2\cdots z_n} = \ol{z}_1\ol{z}_2\cdots\ol{z}_n\) for all \(z_1, z_2, \ldots, z_n\in\C\) (use induction)
  3. \(\ol{(z^n)}=(\ol{z})^n\) for all \(z\in\C\)
  4. Let \(p(z)=a_nz^n+a_{n-1}z^{n-1}+\ldots+a_1z+a_0\) be a polynomial with real coefficients \(a_0,\ldots,a_n\in\R.\) Prove that \(\ol{p(z)} = p({\ol{z}}).\) Deduce that all roots of \(p(z)\) occur in complex conjugate pairs.
Exercise 1.20

Show that \(|z|=|{-z}|\) and \(|\ol z|=|z|.\) Prove also that \(|\la z|=\la |z|\) for all \(\la\geqslant 0.\)

Exercise 1.21

Prove that \(\Re(z)=\frac{1}{2}(z+\bar{z}),\) \(\Im(z)=\frac{1}{2i}(z-\bar{z}).\)

Exercise 1.22

Prove that \(\ol{e^z}=e^{\ol z}.\) Deduce that \(|e^z|=e^{\Re(z)}.\)

Exercise 1.23

Prove that \(|z+w|^2+|z-w|^2=2|z|^2+2|w|^2.\)

Exercise 1.24

Show that \(|z+w|^2=|z|^2+2\Re(z\ol w)+|w|^2.\) Use this to determine the conditions on \(z,w\) for \(|z+w|=|z|+|w|\) to hold.

Exercise 1.25

Assuming we know the triangle inequality \(|z+w|\leqslant |z|+|w|\) for all \(z,w\in\C,\) prove the reverse triangle inequality \[|z-w|\geqslant\bigl||z|-|w|\bigr|.\]

Exercise 1.26

Let \(K\) be a field with \(\R\subset K \subset\C.\) Prove that \(K=\R\) or \(K=\C.\)

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